EPSG Dataset :: v12.003

General polynomial of degree 6

Coordinate Operation Method Details [VALID]

Name:

General polynomial of degree 6

Code:

9648

Operation is Reversible:

Formula:

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as limitations in the transcription will be avoided.

The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system (CRS) to ‘manageable’ numbers, between –10 and +10 at most. This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.

Hence an evaluation point is chosen in the source CRS (XS0, YS0) and in the target CRS (XT0, YT0). Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target CRS. (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).

The selection of an evaluation point in each of the two CRSs allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT – XT0
YT – YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding CRS.)

A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:

U = mS.(XS - XS0)
V = mS.(YS - YS0)

where
XS , YS are coordinates in the source CRS,
XS0 , YS0 are coordinates of the evaluation point in the source CRS,
mS is the scaling factor applied the coordinate differences in the source CRS.

The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.

mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
+ A10.U^4 + A11.U^3.V + A12.U^2.V^2 + A13.U.V^3 + A14.V^4
+ A15.U^5 + A16.U^4.V + A17.U^3.V^2 + A18.U^2.V^3 + A19.U.V^4 + A20.V^5
+ A21.U^6 + A22.U^5.V + A23.U^4.V^2 + A24.U^3.V^3 + A25.U^2.V^4 + A26.U.V^5 + A27.V^6

mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
+ B10.U^4 + B11.U^3.V + B12.U^2.V^2 + B13.U.V^3 + B14.V^4
+ B15.U^5 + B16.U^4.V + B17.U^3.V^2 + B18.U^2.V^3 + B19.U.V^4 + B20.V^5
+ B21.U^6 + B22.U^5.V + B23.U^4.V^2 + B24.U^3.V^3 + B25.U^2.V^4 + B26.U.V^5 + B27.V^6

from which dX and dY are evaluated. These will be in the units of the target CRS.

The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A17 is represented as coordinate operation parameter Au3v2.

The relationship between the two CRSs can now be written as follows:

(XT - XT0) = (XS – XS0) + d
(YT - YT0) = (YS – YS0) + dY
or
XT = XS – XS0 + XT0 + d
YT = YS – YS0 + YT0 + dY

where:
XT, YT are coordinates in the target CRS,
XS, YS are coordinates in the source CRS,
XS0, YS0 are coordinates of the evaluation point in the source CRS,
XT , YT0 are coordinates of the evaluation point in the target CRS,
dX, dY are derived through the scaled polynomial formulas.

Example:

See EPSG Guidance Note 7-2.

Method
Parameters:

Parameter Name	Parameter Code	Parameter Description
Ordinate 1 of evaluation point in source CRS	8619	The value of the first ordinate of the evaluation point expressed in the source coordinate reference system.
Ordinate 2 of evaluation point in source CRS	8620	The value of the second ordinate of the evaluation point expressed in the source coordinate reference system.
Ordinate 1 of evaluation point in target CRS	8621	The value of the first ordinate of the evaluation point expressed in the target coordinate reference system. In the case of an affine transformation the evaluation point is the origin of the source coordinate reference system.
Ordinate 2 of evaluation point in target CRS	8622	The value of the second ordinate of the evaluation point expressed in the target coordinate reference system. In the case of an affine transformation the evaluation point is the origin of the source coordinate reference system.
Scaling factor for source CRS coord differences	8694	Used in general polynomial transformations to normalise coordinate differences to an acceptable numerical range.
Scaling factor for target CRS coord differences	8695	Used in general polynomial transformations to normalise coordinate differences to an acceptable numerical range.
A0	8623	Coefficient used in affine (general parametric) and polynomial transformations.
Au1v0	8716	Coefficient used in polynomial transformations.
Au0v1	8717	Coefficient used in polynomial transformations.
Au2v0	8718	Coefficient used in polynomial transformations.
Au1v1	8719	Coefficient used in polynomial transformations.
Au0v2	8720	Coefficient used in polynomial transformations.
Au3v0	8721	Coefficient used in polynomial transformations.
Au2v1	8722	Coefficient used in polynomial transformations.
Au1v2	8723	Coefficient used in polynomial transformations.
Au0v3	8632	Coefficient used in polynomial transformations.
Au4v0	8633	Coefficient used in polynomial transformations.
Au3v1	8634	Coefficient used in polynomial transformations.
Au2v2	8635	Coefficient used in polynomial transformations.
Au1v3	8636	Coefficient used in polynomial transformations.
Au0v4	8637	Coefficient used in polynomial transformations.
Au5v0	8668	Coefficient used in polynomial transformations.
Au4v1	8669	Coefficient used in polynomial transformations.
Au3v2	8670	Coefficient used in polynomial transformations.
Au2v3	8671	Coefficient used in polynomial transformations.
Au1v4	8672	Coefficient used in polynomial transformations.
Au0v5	8673	Coefficient used in polynomial transformations.
Au6v0	8674	Coefficient used in polynomial transformations.
Au5v1	8675	Coefficient used in polynomial transformations.
Au4v2	8676	Coefficient used in polynomial transformations.
Au3v3	8677	Coefficient used in polynomial transformations.
Au2v4	8678	Coefficient used in polynomial transformations.
Au1v5	8679	Coefficient used in polynomial transformations.
Au0v6	8680	Coefficient used in polynomial transformations.
B0	8639	Coefficient used in affine (general parametric) and polynomial transformations.
Bu1v0	8724	Coefficient used in polynomial transformations.
Bu0v1	8725	Coefficient used in polynomial transformations.
Bu2v0	8726	Coefficient used in polynomial transformations.
Bu1v1	8643	Coefficient used in polynomial transformations.
Bu0v2	8644	Coefficient used in polynomial transformations.
Bu3v0	8645	Coefficient used in polynomial transformations.
Bu2v1	8646	Coefficient used in polynomial transformations.
Bu1v2	8647	Coefficient used in polynomial transformations.
Bu0v3	8648	Coefficient used in polynomial transformations.
Bu4v0	8649	Coefficient used in polynomial transformations.
Bu3v1	8650	Coefficient used in polynomial transformations.
Bu2v2	8651	Coefficient used in polynomial transformations.
Bu1v3	8652	Coefficient used in polynomial transformations.
Bu0v4	8653	Coefficient used in polynomial transformations.
Bu5v0	8681	Coefficient used in polynomial transformations.
Bu4v1	8682	Coefficient used in polynomial transformations.
Bu3v2	8683	Coefficient used in polynomial transformations.
Bu2v3	8684	Coefficient used in polynomial transformations.
Bu1v4	8685	Coefficient used in polynomial transformations.
Bu0v5	8686	Coefficient used in polynomial transformations.
Bu6v0	8687	Coefficient used in polynomial transformations.
Bu5v1	8688	Coefficient used in polynomial transformations.
Bu4v2	8689	Coefficient used in polynomial transformations.
Bu3v3	8690	Coefficient used in polynomial transformations.
Bu2v4	8691	Coefficient used in polynomial transformations.
Bu1v5	8692	Coefficient used in polynomial transformations.
Bu0v6	8693	Coefficient used in polynomial transformations.

Meta Data

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

June 13, 2017

Change ID:

[2017.018]

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